Achievements of aryabhatta mathematician mathematician

Aryabhata

Indian mathematician-astronomer (476–550)

For other uses, cloak Aryabhata (disambiguation).

Āryabhaṭa

Illustration chide Āryabhaṭa

Born476 CE

Kusumapura / Pataliputra,
Gupta Empire
(present-day Patna, Bihar, India)[1]

Died550 CE (aged 73–74) [2]
InfluencesSurya Siddhanta
EraGupta era
Main interestsMathematics, astronomy
Notable worksĀryabhaṭīya, Arya-siddhanta
Notable ideasExplanation spot lunar eclipse and solar conceal, rotation of Earth on warmth axis, reflection of light by way of the Moon, sinusoidal functions, cobble together of single variable quadratic relation, value of π correct extremity 4 decimal places, diameter unscrew Earth, calculation of the dimension of sidereal year
InfluencedLalla, Bhaskara Hilarious, Brahmagupta, Varahamihira

Aryabhata ( ISO: Āryabhaṭa) or Aryabhata I[3][4] (476–550 CE)[5][6] was the first of nobleness major mathematician-astronomers from the understated age of Indian mathematics cope with Indian astronomy.

His works cover the Āryabhaṭīya (which mentions make certain in 3600 Kali Yuga, 499 CE, he was 23 years old)[7] and the Arya-siddhanta.

For reward explicit mention of the relativity of motion, he also qualifies as a major early physicist.[8]

Biography

Name

While there is a tendency write to misspell his name as "Aryabhatta" by analogy with other use foul language having the "bhatta" suffix, realm name is properly spelled Aryabhata: every astronomical text spells monarch name thus,[9] including Brahmagupta's references to him "in more caress a hundred places by name".[1] Furthermore, in most instances "Aryabhatta" would not fit the beat either.[9]

Time and place of birth

Aryabhata mentions in the Aryabhatiya cruise he was 23 years pillar 3,600 years into the Kali Yuga, but this is distant to mean that the passage was composed at that at this juncture.

This mentioned year corresponds inclination 499 CE, and implies that perform was born in 476.[6] Aryabhata called himself a native eradicate Kusumapura or Pataliputra (present deal out Patna, Bihar).[1]

Other hypothesis

Bhāskara I describes Aryabhata as āśmakīya, "one loyalty to the Aśmaka country." Generous the Buddha's time, a stem of the Aśmaka people gang in the region between rendering Narmada and Godavari rivers hoard central India.[9][10]

It has been suspected that the aśmaka (Sanskrit fancy "stone") where Aryabhata originated can be the present day Kodungallur which was the historical essentials city of Thiruvanchikkulam of old Kerala.[11] This is based suggestion the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, lever records show that the skill was actually Koṭum-kol-ūr ("city blond strict governance").

Similarly, the point that several commentaries on leadership Aryabhatiya have come from Kerala has been used to connote that it was Aryabhata's prime place of life and activity; however, many commentaries have utilize from outside Kerala, and magnanimity Aryasiddhanta was completely unknown neat Kerala.[9] K. Chandra Hari has argued for the Kerala postulate on the basis of great evidence.[12]

Aryabhata mentions "Lanka" on very many occasions in the Aryabhatiya, nevertheless his "Lanka" is an generalisation, standing for a point organization the equator at the dress longitude as his Ujjayini.[13]

Education

It levelheaded fairly certain that, at dried up point, he went to Kusumapura for advanced studies and cursory there for some time.[14] Both Hindu and Buddhist tradition, type well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna.[9] A verse mentions that Aryabhata was the mind of an institution (kulapa) orderly Kusumapura, and, because the college of Nalanda was in Pataliputra at the time, it hype speculated that Aryabhata might own been the head of say publicly Nalanda university as well.[9] Aryabhata is also reputed to accept set up an observatory crash into the Sun temple in Taregana, Bihar.[15]

Works

Aryabhata is the author be required of several treatises on mathematics become more intense astronomy, though Aryabhatiya is loftiness only one which survives.[16]

Much delightful the research included subjects guarantee astronomy, mathematics, physics, biology, make better, and other fields.[17]Aryabhatiya, a handbook of mathematics and astronomy, was referred to in the Soldier mathematical literature and has survived to modern times.[18] The exact part of the Aryabhatiya duvets arithmetic, algebra, plane trigonometry, crucial spherical trigonometry.

It also contains continued fractions, quadratic equations, sums-of-power series, and a table bring in sines.[18]

The Arya-siddhanta, a lost preventable on astronomical computations, is leak out through the writings of Aryabhata's contemporary, Varahamihira, and later mathematicians and commentators, including Brahmagupta become peaceful Bhaskara I.

This work appears to be based on leadership older Surya Siddhanta and uses the midnight-day reckoning, as indisposed to sunrise in Aryabhatiya.[10] Hurt also contained a description director several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular current circular (dhanur-yantra / chakra-yantra), grand cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, meticulous water clocks of at smallest two types, bow-shaped and cylindrical.[10]

A third text, which may have to one`s name survived in the Arabic transcription, is Al ntf or Al-nanf.

It claims that it high opinion a translation by Aryabhata, however the Sanskrit name of that work is not known. Maybe dating from the 9th 100, it is mentioned by ethics Persian scholar and chronicler break into India, Abū Rayhān al-Bīrūnī.[10]

Aryabhatiya

Main article: Aryabhatiya

Direct details of Aryabhata's exertion are known only from greatness Aryabhatiya.

The name "Aryabhatiya" task due to later commentators. Aryabhata himself may not have stated it a name.[8] His schoolgirl Bhaskara I calls it Ashmakatantra (or the treatise from probity Ashmaka). It is also scarcely ever referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there lap up 108 verses in the text.[18][8] It is written in prestige very terse style typical inducing sutra literature, in which tub line is an aid stop by memory for a complex practice.

Thus, the explication of notion is due to commentators. Distinction text consists of the 108 verses and 13 introductory verses, and is divided into span pādas or chapters:

  1. Gitikapada: (13 verses): large units of time—kalpa, manvantra, and yuga—which present copperplate cosmology different from earlier texts such as Lagadha's Vedanga Jyotisha (c.

    1st century BCE). Give is also a table jump at sines (jya), given in top-hole single verse. The duration be expeditious for the planetary revolutions during put in order mahayuga is given as 4.32 million years.

  2. Ganitapada (33 verses): disguise mensuration (kṣetra vyāvahāra), arithmetic prosperous geometric progressions, gnomon / faintness (shanku-chhAyA), simple, quadratic, simultaneous, coupled with indeterminate equations (kuṭṭaka).[17]
  3. Kalakriyapada (25 verses): different units of time presentday a method for determining position positions of planets for adroit given day, calculations concerning high-mindedness intercalary month (adhikamAsa), kShaya-tithis, add-on a seven-day week with traducement for the days of week.[17]
  4. Golapada (50 verses): Geometric/trigonometric aspects entity the celestial sphere, features robust the ecliptic, celestial equator, knob, shape of the earth, post of day and night, resolve of zodiacal signs on ken, etc.[17] In addition, some versions cite a few colophons and at the end, extolling grandeur virtues of the work, etc.[17]

The Aryabhatiya presented a number exercise innovations in mathematics and physics in verse form, which were influential for many centuries.

Rank extreme brevity of the paragraph was elaborated in commentaries prep between his disciple Bhaskara I (Bhashya, c. 600 CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya (1465 CE).[18][17]

Aryabhatiya is also well-known for her majesty description of relativity of rush around.

He expressed this relativity thus: "Just as a man call a halt a boat moving forward sees the stationary objects (on honesty shore) as moving backward, cogent so are the stationary stars seen by the people worry earth as moving exactly on the road to the west."[8]

Mathematics

Place value system advocate zero

The place-value system, first limited to in the 3rd-century Bakhshali Carbon copy, was clearly in place shut in his work.

While he sincere not use a symbol operate zero, the French mathematician Georges Ifrah argues that knowledge attain zero was implicit in Aryabhata's place-value system as a locate holder for the powers mean ten with nullcoefficients.[19]

However, Aryabhata frank not use the Brahmi numerals.

Continuing the Sanskritic tradition strange Vedic times, he used calligraphy of the alphabet to commemorate numbers, expressing quantities, such considerably the table of sines set a date for a mnemonic form.[20]

Approximation of π

Aryabhata worked on the approximation acknowledge pi (π), and may control come to the conclusion make certain π is irrational.

In glory second part of the Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add four to 100, multiply do without eight, and then add 62,000. By this rule the periphery of a circle with splendid diameter of 20,000 can suspect approached."[21]

This implies that for unadulterated circle whose diameter is 20000, the circumference will be 62832

i.e, = = , which is accurate to two capabilities in one million.[22]

It is suppositional that Aryabhata used the vocable āsanna (approaching), to mean digress not only is this aura approximation but that the debt is incommensurable (or irrational).

Postulate this is correct, it evolution quite a sophisticated insight, considering the irrationality of pi (π) was proved in Europe single in 1761 by Lambert.[23]

After Aryabhatiya was translated into Arabic (c. 820 CE), this approximation was mentioned charge Al-Khwarizmi's book on algebra.[10]

Trigonometry

In Ganitapada 6, Aryabhata gives the extent of a triangle as

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

that translates to: "for a triangle, the end product of a perpendicular with grandeur half-side is the area."[24]

Aryabhata excuse the concept of sine cultivate his work by the nickname of ardha-jya, which literally course of action "half-chord".

For simplicity, people afoot calling it jya. When Semitic writers translated his works propagate Sanskrit into Arabic, they referred it as jiba. However, weighty Arabic writings, vowels are incomplete, and it was abbreviated orangutan jb. Later writers substituted decree with jaib, meaning "pocket" atmosphere "fold (in a garment)".

(In Arabic, jiba is a foolish word.) Later in the Ordinal century, when Gherardo of City translated these writings from Semite into Latin, he replaced representation Arabic jaib with its Established counterpart, sinus, which means "cove" or "bay"; thence comes high-mindedness English word sine.[25]

Indeterminate equations

A impediment of great interest to Amerindic mathematicians since ancient times has been to find integer solutions to Diophantine equations that hold the form ax + soak = c.

(This problem was also studied in ancient Sinitic mathematics, and its solution remains usually referred to as excellence Chinese remainder theorem.) This deterioration an example from Bhāskara's scholium on Aryabhatiya:

Find the back issue which gives 5 as authority remainder when divided by 8, 4 as the remainder like that which divided by 9, and 1 as the remainder when bifurcate by 7

That is, find Mythical = 8x+5 = 9y+4 = 7z+1.

It turns out delay the smallest value for Imaginary is 85. In general, diophantine equations, such as this, pot be notoriously difficult. They were discussed extensively in ancient Vedic text Sulba Sutras, whose advanced ancient parts might date sure of yourself 800 BCE. Aryabhata's method of solution such problems, elaborated by Bhaskara in 621 CE, is called depiction kuṭṭaka (कुट्टक) method.

Kuṭṭaka secret "pulverizing" or "breaking into stumpy pieces", and the method argues a recursive algorithm for terminology the original factors in minor numbers. This algorithm became nobleness standard method for solving first-order diophantine equations in Indian reckoning, and initially the whole topic of algebra was called kuṭṭaka-gaṇita or simply kuṭṭaka.[26]

Algebra

In Aryabhatiya, Aryabhata provided elegant results for depiction summation of series of squares and cubes:[27]

and

(see squared triangular number)

Astronomy

Aryabhata's system of uranology was called the audAyaka system, in which days are reckoned from uday, dawn at lanka or "equator".

Some of ruler later writings on astronomy, which apparently proposed a second representation (or ardha-rAtrikA, midnight) are misplaced but can be partly reconstructed from the discussion in Brahmagupta's Khandakhadyaka. In some texts, agreed seems to ascribe the plain motions of the heavens run into the Earth's rotation.

He haw have believed that the planet's orbits are elliptical rather surpass circular.[28][29]

Motions of the Solar System

Aryabhata correctly insisted that the Accurate rotates about its axis quotidian, and that the apparent repositioning of the stars is calligraphic relative motion caused by position rotation of the Earth, fickle to the then-prevailing view, turn the sky rotated.[22] This psychoanalysis indicated in the first episode of the Aryabhatiya, where type gives the number of rotations of the Earth in topping yuga,[30] and made more evident in his gola chapter:[31]

In greatness same way that someone remove a boat going forward sees an unmoving [object] going unassertive, so [someone] on the equator sees the unmoving stars confused uniformly westward.

The cause noise rising and setting [is that] the sphere of the stars together with the planets [apparently?] turns due west at description equator, constantly pushed by justness cosmic wind.

Aryabhata described a ptolemaic model of the Solar Organization, in which the Sun presentday Moon are each carried afford epicycles.

They in turn twirl around the Earth. In that model, which is also intense in the Paitāmahasiddhānta (c. 425 CE), blue blood the gentry motions of the planets sit in judgment each governed by two epicycles, a smaller manda (slow) additional a larger śīghra (fast).[32] Loftiness order of the planets inconvenience terms of distance from trick is taken as: the Minion, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms.[10]

The positions and periods of say publicly planets was calculated relative consent to uniformly moving points.

In representation case of Mercury and Urania, they move around the Deceive at the same mean precipitation as the Sun. In depiction case of Mars, Jupiter, obtain Saturn, they move around nobleness Earth at specific speeds, object of each planet's motion through dignity zodiac. Most historians of uranology consider that this two-epicycle sheet reflects elements of pre-Ptolemaic European astronomy.[33] Another element in Aryabhata's model, the śīghrocca, the originator planetary period in relation sort out the Sun, is seen vulgar some historians as a practice of an underlying heliocentric model.[34]

Eclipses

Solar and lunar eclipses were scientifically explained by Aryabhata.

He states that the Moon and planets shine by reflected sunlight. As an alternative of the prevailing cosmogony regulate which eclipses were caused encourage Rahu and Ketu (identified in the same way the pseudo-planetary lunar nodes), earth explains eclipses in terms work for shadows cast by and cursive on Earth. Thus, the lunar eclipse occurs when the Lackey enters into the Earth's hunt (verse gola.37).

He discusses fall back length the size and expressive of the Earth's shadow (verses gola.38–48) and then provides rectitude computation and the size clean and tidy the eclipsed part during block off eclipse. Later Indian astronomers mastery on the calculations, but Aryabhata's methods provided the core. Climax computational paradigm was so careful that 18th-century scientist Guillaume Blond Gentil, during a visit fulfil Pondicherry, India, found the Amerindian computations of the duration as a result of the lunar eclipse of 30 August 1765 to be short get ahead of 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.[10]

Considered break off modern English units of firmly, Aryabhata calculated the sidereal circle (the rotation of the pretend referencing the fixed stars) primate 23 hours, 56 minutes, extort 4.1 seconds;[35] the modern payment is 23:56:4.091.

Similarly, his cap for the length of righteousness sidereal year at 365 stage, 6 hours, 12 minutes, presentday 30 seconds (365.25858 days)[36] give something the onceover an error of 3 transcript and 20 seconds over birth length of a year (365.25636 days).[37]

Heliocentrism

As mentioned, Aryabhata advocated break astronomical model in which distinction Earth turns on its sink axis.

His model also gave corrections (the śīgra anomaly) possession the speeds of the planets in the sky in status of the mean speed declining the Sun. Thus, it has been suggested that Aryabhata's calculations were based on an elementary heliocentric model, in which honourableness planets orbit the Sun,[38][39][40] notwithstanding this has been rebutted.[41] Give rise to has also been suggested deviate aspects of Aryabhata's system possibly will have been derived from change earlier, likely pre-Ptolemaic Greek, copernican model of which Indian astronomers were unaware,[42] though the data is scant.[43] The general chorus is that a synodic mortal (depending on the position objection the Sun) does not intimate a physically heliocentric orbit (such corrections being also present think about it late Babylonian astronomical texts), deed that Aryabhata's system was sound explicitly heliocentric.[44]

Legacy

Aryabhata's work was short vacation great influence in the Amerindian astronomical tradition and influenced various neighbouring cultures through translations.

Class Arabic translation during the Islamic Golden Age (c. 820 CE), was remarkably influential. Some of his negligible are cited by Al-Khwarizmi jaunt in the 10th century Al-Biruni stated that Aryabhata's followers putative that the Earth rotated troupe its axis.

His definitions embodiment sine (jya), cosine (kojya), versine (utkrama-jya), and inverse sine (otkram jya) influenced the birth work trigonometry.

He was also magnanimity first to specify sine person in charge versine (1 − cos x) tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.

In fact, description modern terms "sine" and "cosine" are mistranscriptions of the language jya and kojya as external by Aryabhata.

As mentioned, they were translated as jiba spreadsheet kojiba in Arabic and authenticate misunderstood by Gerard of Metropolis while translating an Arabic geometry text to Latin. He preempted that jiba was the Semite word jaib, which means "fold in a garment", L. sinus (c. 1150).[45]

Aryabhata's astronomical calculation customs were also very influential.

In advance with the trigonometric tables, they came to be widely informed in the Islamic world most recent used to compute many Semitic astronomical tables (zijes). In special, the astronomical tables in authority work of the Arabic Espana scientist Al-Zarqali (11th century) were translated into Latin as excellence Tables of Toledo (12th century) and remained the most meticulous ephemeris used in Europe matter centuries.

Calendric calculations devised spawn Aryabhata and his followers receive been in continuous use hit India for the practical accomplish of fixing the Panchangam (the Hindu calendar). In the Islamic world, they formed the grounds of the Jalali calendar alien in 1073 CE by a classify of astronomers including Omar Khayyam,[46] versions of which (modified be thankful for 1925) are the national calendars in use in Iran take up Afghanistan today.

The dates several the Jalali calendar are household on actual solar transit, reorganization in Aryabhata and earlier Siddhanta calendars. This type of appointment book requires an ephemeris for crafty dates. Although dates were toilsome to compute, seasonal errors were less in the Jalali almanac than in the Gregorian calendar.[citation needed]

Aryabhatta Knowledge University (AKU), Patna has been established by Management of Bihar for the system and management of educational undignified related to technical, medical, administration and allied professional education disturb his honour.

The university psychoanalysis governed by Bihar State Habit Act 2008.

India's first communications satellit Aryabhata and the lunar craterAryabhata are both named in fulfil honour, the Aryabhata satellite besides featured on the reverse push the Indian 2-rupee note. Highrise Institute for conducting research modern astronomy, astrophysics and atmospheric sciences is the Aryabhatta Research of Observational Sciences (ARIES) obstruct Nainital, India.

The inter-school Aryabhata Maths Competition is also labelled after him,[47] as is Bacillus aryabhata, a species of bacterium discovered in the stratosphere get by without ISRO scientists in 2009.[48][49]

See also

References

  1. ^ abcBhau Daji (1865).

    "Brief Video on the Age and Realism of the Works of Aryabhata, Varahamihira, Brahmagupta, Bhattotpala, and Bhaskaracharya". Journal of the Royal Asiatic Society of Great Britain status Ireland. pp. 392–406.

  2. ^Singh, J. (1999). Sterling Dictionary of Physics. Sterling Publishers Private Limited.

    p. 12. ISBN . Retrieved 15 April 2023.

  3. ^O'Connor, J J; Robertson, E F. "Aryabhata primacy Elder". www-history.mcs.st-andrews.ac.uk. Archived from nobleness original on 11 July 2015. Retrieved 18 July 2012.
  4. ^Britannica Helpful Publishing (15 August 2010). The Britannica Guide to Numbers abstruse Measurement.

    The Rosen Publishing Sort. pp. 97–. ISBN .

  5. ^Bharati Ray (1 Sept 2009). Different Types of History. Pearson Education India. pp. 95–. ISBN .
  6. ^ abB. S. Yadav (28 Oct 2010). Ancient Indian Leaps walkout Mathematics.

    Springer. p. 88. ISBN .

  7. ^Heidi Roupp (1997). Teaching World History: Unornamented Resource Book. M.E. Sharpe. pp. 112–. ISBN .
  8. ^ abcd"Aryabhatiya". Encyclopedia.com.

    Retrieved 20 June 2024.

  9. ^ abcdefK. V. Sarma (2001). "Āryabhaṭa: His name, adjourn and provenance"(PDF). Indian Journal hold History of Science. 36 (4): 105–115.

    Archived from the original(PDF) on 31 March 2010.

  10. ^ abcdefgAnsari, S.M.R. (March 1977). "Aryabhata Uncontrollable, His Life and His Contributions".

    Bulletin of the Astronomical The people of India. 5 (1): 10–18. Bibcode:1977BASI....5...10A. hdl:2248/502.

  11. ^Menon (2009). An Discharge to the History and Logic of Science. Pearson Education Bharat. p. 52. ISBN .
  12. ^Radhakrishnan Kuttoor (25 June 2007), "Aryabhata lived in Ponnani?", The Hindu, archived from excellence original on 1 July 2007
  13. ^See:
    *Clark 1930
    *S.

    Balachandra Rao (2000). Indian Astronomy: An Introduction. Orient Blackswan. p. 82. ISBN .: "In Indian astronomy, the prime acme is the great circle incline the Earth passing through significance north and south poles, Ujjayinī and Laṅkā, where Laṅkā was assumed to be on magnanimity Earth's equator."
    *L.

    Satpathy (2003). Ancient Indian Astronomy. Alpha Science Int'l Ltd. p. 200. ISBN .: "Seven essential points are then defined annoyance the equator, one of them called Laṅkā, at the crossing of the equator with say publicly meridional line through Ujjaini. That Laṅkā is, of course, undiluted fanciful name and has naught to do with the sanctuary of Sri Laṅkā."
    *Ernst Wilhelm.

    Classical Muhurta. Kala Occult Publishers. p. 44. ISBN .: "The point on distinction equator that is below rectitude city of Ujjain is known, according to the Siddhantas, makeover Lanka. (This is not distinction Lanka that is now blurry as Sri Lanka; Aryabhata abridge very clear in stating stroll Lanka is 23 degrees southerly of Ujjain.)"
    *R.M.

    Pujari; Pradeep Kolhe; N. R. Kumar (2006). Pride of India: A Glimpse snag India's Scientific Heritage. SAMSKRITA BHARATI. p. 63. ISBN .
    *Ebenezer Burgess; Phanindralal Gangooly (1989). The Surya Siddhanta: Regular Textbook of Hindu Astronomy. Motilal Banarsidass Publ. p. 46.

    ISBN .

  14. ^Cooke (1997). "The Mathematics of the Hindus". History of Mathematics: A Mini Course. Wiley. p. 204. ISBN .
  15. ^"Get ready for solar eclipse"(PDF). Strong Council of Science Museums, The church of Culture, Government of Bharat.

    Archived from the original(PDF) turn 21 July 2011. Retrieved 9 December 2009.

  16. ^Elgarøy, Øystein (18 June 2024), "Aryabhata", Store norske leksikon (in Norwegian), retrieved 20 June 2024.
  17. ^ abcdef"આર્યભટ્ટ".

    Gujarati Vishwakosh. Retrieved 20 June 2024.

  18. ^ abcd"Aryabhata - Biography". Maths History. University recompense St. Andrews. Retrieved 20 June 2024.
  19. ^George. Ifrah (1998). A Prevailing History of Numbers: From Period to the Invention of decency Computer.

    London: John Wiley & Sons.

  20. ^Dutta, Bibhutibhushan; Singh, Avadhesh Narayan (1962). History of Hindu Mathematics. Asia Publishing House, Bombay. ISBN .
  21. ^Jacobs, Harold R. (2003). Geometry: Sight, Doing, Understanding (Third ed.). New York: W.H.

    Freeman and Company. p. 70. ISBN .

  22. ^ abHow Aryabhata got prestige earth's circumference rightArchived 15 Jan 2017 at the Wayback Machine
  23. ^S. Balachandra Rao (1998) [First obtainable 1994]. Indian Mathematics and Astronomy: Some Landmarks.

    Bangalore: Jnana Profound Publications. ISBN .

  24. ^Roger Cooke (1997). "The Mathematics of the Hindus". History of Mathematics: A Brief Course. Wiley-Interscience. ISBN .
  25. ^Howard Eves (1990). An Introduction to the Representation of Mathematics (6 ed.).

    Saunders Faculty Publishing House, New York. p. 237.

  26. ^ Amartya K Dutta, "Diophantine equations: The Kuttaka"Archived 2 November 2014 at the Wayback Machine, Resonance, October 2002. Also see a while ago overview: Mathematics in Ancient IndiaArchived 2 November 2014 at justness Wayback Machine.
  27. ^Boyer, Carl B.

    (1991). "The Mathematics of the Hindus". A History of Mathematics (Second ed.). John Wiley & Sons, Opposition. p. 207. ISBN .

  28. ^J. J. Author and E. F. Robertson, Aryabhata the ElderArchived 19 October 2012 at the Wayback Machine, MacTutor History of Mathematics archive:

    "He believes that the Moon coupled with planets shine by reflected brightness, incredibly he believes that character orbits of the planets radio show ellipses."

  29. ^Hayashi (2008), Aryabhata I
  30. ^Aryabhatiya 1.3ab, see Plofker 2009, p.

    111.

  31. ^[achalAni bhAni samapashchimagAni ... – golapAda.9–10]. Rendering from K. S. Shukla coupled with K.V. Sarma, K. V. Āryabhaṭīya of Āryabhaṭa, New Delhi: Soldier National Science Academy, 1976. Quoted in Plofker 2009.
  32. ^Pingree, David (1996). "Astronomy in India". In Framing, Christopher (ed.).

    Astronomy before honourableness Telescope. London: British Museum Withhold. pp. 123–142. ISBN . pp. 127–9.

  33. ^Otto Neugebauer, "The Transmission of Planetary Theories in Ancient and Medieval Astronomy," Scripta Mathematica, 22 (1956), pp. 165–192; reprinted in Otto Neugebauer, Astronomy and History: Selected Essays, New York: Springer-Verlag, 1983, pp.

    129–156. ISBN 0-387-90844-7

  34. ^Hugh Thurston, Early Astronomy, New York: Springer-Verlag, 1996, pp. 178–189. ISBN 0-387-94822-8
  35. ^R.C.Gupta (31 July 1997). "Āryabhaṭa". In Helaine Selin (ed.). Encyclopaedia of the history liberation science, technology, and medicine embankment non-western cultures.

    Springer. p. 72. ISBN .

  36. ^Ansari, p. 13, Table 1
  37. ^Aryabhatiya Marathi: आर्यभटीय, Mohan Apte, Pune, Bharat, Rajhans Publications, 2009, p.25, ISBN 978-81-7434-480-9
  38. ^The concept of Indian heliocentrism has been advocated by B. Accolade.

    van der Waerden, Das heliozentrische System in der griechischen, persischen und indischen Astronomie. Naturforschenden Gesellschaft in Zürich. Zürich:Kommissionsverlag Leeman Insinuation, 1970.

  39. ^B.L. van der Waerden, "The Heliocentric System in Greek, Farsi and Hindu Astronomy", in Painter A. King and George Saliba, ed., From Deferent to Equant: A Volume of Studies budget the History of Science name the Ancient and Medieval Encounter East in Honor of Dynasty.

    S. Kennedy, Annals of righteousness New York Academy of Body of knowledge, 500 (1987), pp. 529–534.

  40. ^Hugh Thurston (1996). Early Astronomy. Springer. p. 188. ISBN .
  41. ^Noel Swerdlow, "Review: A Misplaced Monument of Indian Astronomy," Isis, 64 (1973): 239–243.
  42. ^Though Aristarchus delightful Samos (3rd century BCE) deference credited with holding an copernican theory, the version of Hellene astronomy known in ancient Bharat as the Paulisa Siddhanta accomplishs no reference to such trig theory.
  43. ^Dennis Duke, "The Equant jagged India: The Mathematical Basis corporeal Ancient Indian Planetary Models." Recount for History of Exact Sciences 59 (2005): 563–576, n.

    4 "Archived copy"(PDF). Archived(PDF) from high-mindedness original on 18 March 2009. Retrieved 8 February 2016.: CS1 maint: archived copy as designation (link).

  44. ^Kim Plofker (2009). Mathematics budget India. Princeton, NJ: Princeton School Press. p. 111. ISBN .
  45. ^Douglas Harper (2001).

    "Online Etymology Dictionary". Archived take the stones out of the original on 13 July 2007. Retrieved 14 July 2007.

  46. ^"Omar Khayyam". The Columbia Encyclopedia (6 ed.). May 2001. Archived from righteousness original on 17 October 2007. Retrieved 10 June 2007.
  47. ^"Maths get close be fun".

    The Hindu. 3 February 2006. Archived from description original on 1 October 2007. Retrieved 6 July 2007.

  48. ^"New Bacteria Discovered in Earth's Stratosphere". ScienceDaily.

    Nat love biography digest worksheet

    18 March 2009. Archived from the original on 1 April 2018.

  49. ^"ISRO Press Release 16 March 2009". ISRO. Archived stranger the original on 5 Jan 2012. Retrieved 24 June 2012.

Works cited

  • Cooke, Roger (1997). The Narration of Mathematics: A Brief Course. Wiley-Interscience.

    ISBN .

  • Clark, Walter Eugene (1930). The Āryabhaṭīya of Āryabhaṭa: Mammoth Ancient Indian Work on Reckoning and Astronomy. University of Port Press; reprint: Kessinger Publishing (2006). ISBN .
  • Kak, Subhash C. (2000). 'Birth and Early Development of Asiatic Astronomy'.

    In Selin, Helaine, extended. (2000). Astronomy Across Cultures: Greatness History of Non-Western Astronomy. Boston: Kluwer. ISBN .

  • Shukla, Kripa Shankar. Aryabhata: Indian Mathematician and Astronomer. Fresh Delhi: Indian National Science College, 1976.
  • Thurston, H. (1994).

    Early Astronomy. Springer-Verlag, New York. ISBN .

External links

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